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r"""
This model calculates the scattering from fractal-like aggregates based
on the Mildner reference.
Definition
----------
The scattering intensity $I(q)$ is calculated as
.. math::
:nowrap:
\begin{align*}
I(q) &= \text{scale} \times P(q)S(q) + \text{background} \\
P(q) &= F(qR)^2 \\
F(x) &= \frac{3\left[\sin(x)-x\cos(x)\right]}{x^3} \\
S(q) &= \Gamma(5-D_S)\xi^{\,5-D_S}\left[1+(q\xi)^2 \right]^{-(5-D_S)/2}
\sin\left[-(5-D_S) \tan^{-1}(q\xi) \right] q^{-1} \\
\text{scale} &= \text{scale factor}\,
N V^1(\rho_\text{particle} - \rho_\text{solvent})^2 \\
V &= \frac{4}{3}\pi R^3
\end{align*}
where $R$ is the radius of the building block, $D_S$ is the **surface** fractal
dimension, $\xi$ is the cut-off length, $\rho_\text{solvent}$ is the scattering
length density of the solvent and $\rho_\text{particle}$ is the scattering
length density of particles.
.. note::
The surface fractal dimension is only valid if $1<D_S<3$. The result is only
valid over a limited $q$ range, $\tfrac{5}{3-D_S}\xi^{\,-1} < q < R^{-1}$.
See the reference for details.
References
----------
#. D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf
name = "surface_fractal"
title = "Fractal-like aggregates based on the Mildner reference"
description = """\
[The scattering intensity I(x) = scale*P(x)*S(x) + background, where
scale = scale_factor * V * delta^(2)
p(x) = F(x*radius)^(2)
F(x) = 3*[sin(x)-x cos(x)]/x**3
S(x) = [(gamma(5-Ds)*colength^(5-Ds)*[1+(x^2*colength^2)]^((Ds-5)/2)
* sin[(Ds-5)*arctan(x*colength)])/x]
where
delta = sldParticle -sldSolv.
radius = Particle radius
fractal_dim_surf = Surface fractal dimension (Ds)
co_length = Cut-off length
background = background
Ref. :Mildner, Hall,J Phys D Appl Phys(1986), 19, 1535-1545
Note I : This model is valid for 1<fractal_dim_surf<3 with limited q range.
Note II: This model is not in absolute scale.
"""
category = "shape-independent"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["radius", "Ang", 10.0, [0, inf], "",
"Particle radius"],
["fractal_dim_surf", "", 2.0, [1, 3], "",
"Surface fractal dimension"],
["cutoff_length", "Ang", 500., [0.0, inf], "",
"Cut-off Length"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "surface_fractal.c"]
# Don't need validity test since fractal_dim_surf is not polydisperse
#valid = "fractal_dim_surf > 1.0 && fractal_dim_surf < 3.0"
def random():
"""Return a random parameter set for the model."""
radius = 10**np.random.uniform(1, 4)
fractal_dim_surf = np.random.uniform(1, 3-1e-6)
cutoff_length = 1e6 # Sets the low q limit; keep it big for sim
pars = dict(
#background=0,
scale=1,
radius=radius,
fractal_dim_surf=fractal_dim_surf,
cutoff_length=cutoff_length,
)
return pars
tests = [
# Accuracy tests based on content in test/utest_other_models.py
[{'radius': 10.0,
'fractal_dim_surf': 2.0,
'cutoff_length': 500.0,
}, 0.05, 301428.66016],
# Additional tests with larger range of parameters
[{'radius': 1.0,
'fractal_dim_surf': 1.0,
'cutoff_length': 10.0,
}, 0.332070182643, 1125.00421004],
[{'radius': 3.5,
'fractal_dim_surf': 0.1,
'cutoff_length': 30.0,
'background': 0.01,
}, 5.0, 0.00999998891322],
[{'radius': 3.0,
'fractal_dim_surf': 1.0,
'cutoff_length': 33.0,
'scale': 0.1,
}, 0.51, 2.50120147004],
]
|
